Solving For Two Numbers Given Their Sum And A Linear Combination

In mathematics, we often encounter problems that require us to find unknown numbers based on given conditions. These problems can range from simple to complex, but they all share a common thread: the need to translate word problems into mathematical equations and then solve those equations. This article will delve into a specific type of problem involving two numbers and two conditions, demonstrating a systematic approach to finding the solution. We will explore how to set up a system of equations, solve it using different methods, and interpret the results. By understanding these techniques, you'll be well-equipped to tackle similar problems in various mathematical contexts.

Problem Statement: Decoding the Numerical Puzzle

Let's begin by stating the problem we aim to solve: The sum of two numbers is 13. Two times the first number minus three times the second number is 1. If we let $x$ represent the first number and $y$ represent the second number, our goal is to determine the values of $x$ and $y$. This problem is a classic example of a system of linear equations, where we have two unknowns and two equations that relate them. The challenge lies in translating the word problem into mathematical expressions and then employing appropriate methods to solve for the unknowns. The ability to translate real-world scenarios into mathematical models is a fundamental skill in mathematics and has applications in various fields, including science, engineering, and economics.

Translating Words into Equations: The Foundation of the Solution

The first step in solving this problem is to translate the given information into mathematical equations. This process involves carefully identifying the key relationships between the numbers and expressing them using algebraic symbols. The statement "The sum of two numbers is 13" can be directly translated into the equation: $x + y = 13$. This equation represents the first condition given in the problem. Similarly, the statement "Two times the first number minus three times the second number is 1" can be translated into the equation: $2x - 3y = 1$. This equation represents the second condition. We now have a system of two linear equations with two unknowns:

1. $x + y = 13$
2. $2x - 3y = 1$

This system of equations forms the foundation for solving the problem. The next step involves choosing an appropriate method to solve this system and find the values of $x$ and $y$.

Solving the System of Equations: Unveiling the Numbers

There are several methods to solve a system of linear equations, including substitution, elimination, and graphical methods. For this particular problem, we will demonstrate two common methods: substitution and elimination. Each method offers a different approach to isolating the variables and finding their values. Understanding these methods is crucial for developing problem-solving skills and choosing the most efficient method for a given situation.

Method 1: Substitution - A Step-by-Step Approach

The substitution method involves solving one equation for one variable and then substituting that expression into the other equation. This process reduces the system to a single equation with one unknown, which can then be easily solved. Let's apply this method to our system of equations:

1. $x + y = 13$
2. $2x - 3y = 1$

First, we solve equation (1) for $x$:

$x = 13 - y$

Now, we substitute this expression for $x$ into equation (2):

$2(13 - y) - 3y = 1$

Next, we simplify and solve for $y$:

$26 - 2y - 3y = 1$

$-5y = -25$

$y = 5$

Now that we have found the value of $y$, we can substitute it back into the equation $x = 13 - y$ to find the value of $x$:

$x = 13 - 5$

$x = 8$

Therefore, using the substitution method, we find that $x = 8$ and $y = 5$.

Method 2: Elimination - A Strategic Simplification

The elimination method involves manipulating the equations in the system so that the coefficients of one variable are opposites. When the equations are added together, that variable is eliminated, leaving a single equation with one unknown. Let's apply this method to our system of equations:

1. $x + y = 13$
2. $2x - 3y = 1$

To eliminate $x$, we can multiply equation (1) by -2:

$-2(x + y) = -2(13)$

$-2x - 2y = -26$

Now we have the following system:

1. $-2x - 2y = -26$
2. $2x - 3y = 1$

Adding the two equations together eliminates $x$:

(-2x - 2y) + (2x - 3y) = -26 + 1$

$-5y = -25$

$y = 5$

Now that we have found the value of $y$, we can substitute it back into either equation (1) or (2) to find the value of $x$. Let's use equation (1):

$x + 5 = 13$

$x = 8$

Therefore, using the elimination method, we also find that $x = 8$ and $y = 5$. The consistency of the results from both methods reinforces the accuracy of our solution.

Verifying the Solution: Ensuring Accuracy

Once we have found the values of $x$ and $y$, it is crucial to verify that they satisfy both equations in the original system. This step ensures that our solution is accurate and that we have not made any errors in our calculations. Let's verify our solution, $x = 8$ and $y = 5$, using the original equations:

1. $x + y = 13$

   Substituting $x = 8$ and $y = 5$:

   $8 + 5 = 13$

   $13 = 13$ (This equation is satisfied)

2. $2x - 3y = 1$

   Substituting $x = 8$ and $y = 5$:

   $2(8) - 3(5) = 1$

   $16 - 15 = 1$

   $1 = 1$ (This equation is satisfied)

Since our values for $x$ and $y$ satisfy both equations, we can confidently conclude that our solution is correct. This verification process is an essential part of problem-solving and helps to build confidence in the results.

Conclusion: The Numbers Revealed

In conclusion, the two numbers that satisfy the given conditions are $x = 8$ and $y = 5$. We arrived at this solution by translating the word problem into a system of two linear equations and then solving the system using both the substitution and elimination methods. The consistency of the results from both methods, along with the verification step, confirms the accuracy of our solution. This problem demonstrates the power of algebraic techniques in solving real-world problems. By understanding the principles of setting up and solving systems of equations, you can tackle a wide range of mathematical challenges and apply these skills in various practical situations. Remember, the key to success in mathematics lies in a systematic approach, careful attention to detail, and a thorough understanding of the underlying concepts. The ability to translate word problems into mathematical expressions is a valuable skill that will serve you well in your mathematical journey.